$$L_1$$ L 1 splitting rules in survival forests

Abstract

The log-rank test is used as the split function in many commonly used survival trees and forests algorithms. However, the log-rank test may have a significant loss of power in some circumstances, especially when the hazard functions or when the survival functions cross each other in the two compared groups. We investigate the use of the integrated absolute difference between the two children nodes survival functions as the splitting rule. Simulations studies and applications to real data sets show that forests built with this rule produce very good results in general, and that they are often better compared to forests built with the log-rank splitting rule.

Survival forests under test: Impact of the proportional hazards assumption on prognostic and predictive forests for amyotrophic lateral sclerosis survival

“…The recently proposed L 1 -splitting survival forests (Moradian et al 2017) implement splits maximising the integrated absolute difference between two survival functions, where the corresponding groups are defined by a potential binary split. The method does not fit into the theoretical framework discussed here but was designed to deal with non-proportional hazards and thus we compare it empirically to the remaining forest variants in the next Section.…”

“…We compared the seven prognostic models from the prognostic part of Table 1 and, in addition, survival forests based on L 1 splitting (Moradian et al 2017). For all competitors except Ranger, a common set of parameters was specified: 250 trees of maximal depth 10 and not less than 20 observations in a terminal node.…”

“…Using the flexible transformation forests framework (Hothorn and Zeileis 2017), we design novel split criteria for prognostic and predictive survival trees, which are powerful in both the proportional and the non-proportional hazards setting. We compared several variants of survival forests in this class to established survival forests (RF-S, CForest, Ranger) and to one recent proposal (explicitly targeting the non-proportional hazards situation using L 1 splitting, Moradian et al 2017) in an artificial prognostic model setting and then investigated the empirical performance for ALS survival prognosis. In a second step, we compared predictive survival forest models based on Weibull models ("distributional survival forests", DSF, Seibold et al 2018) with a less restrictive novel variant of prognostic and predictive survival forests introduced herein ("transformation survival forests", TSF), both with respect to predictive ALS models and based on simulations.…”

Clinical risk prediction with random forests for survival, longitudinal, and multivariate (RF-SLAM) data analysis

“…Additionally, since the split rule is based upon Poisson regression rather than the log-rank statistic, the splitting does not depend upon the proportional hazards assumption, which is often inappropriate or an oversimplification in the analysis of real life data. Because RSF employs the log-rank statistic for its split rule, it is possible that RSF will be unable to select potentially beneficial splits if the proportional hazards assumption is violated since the key requirement for the log-rank test optimality is proportional hazards [43, 44, 65–68].…”

“…heart failure exacerbations) and the interpretability of RSF predictions in the case of time-dependent outcome data. Additionally, the recent literature has expressed concerns regarding the log-rank split statistic since this is based upon the proportional hazards assumption and may suffer from significant loss of power in situations in which covariates violate the proportional hazards assumption, especially when the hazard/survival functions cross for the groups being compared [43, 44]. As a result, we introduce an extension of the random forest methodology, which we call RF-SLAM, based upon the Poisson regression log-likelihood as the split statistic to allow for the inclusion of time-varying predictors and the analysis of survival data without the restrictive proportional hazards assumption.…”